Abstract Algebra Answers

Let G = [x] be a cyclic group of order 5. Show that a = 2x4 − x3 − 3x2 − x + 2is a unit of infinite order in ZG, with inverse b = 2x4 − 3x3 +2x2 − x − 1.

Let G = [x] be a cyclic group of order 5. Show that U(ZG) = [u] × (±G) ∼ Z ⊕ Z2 ⊕ Z5.

Let G = [x] be a cyclic group of order 5. Show that u = 1− x2 − x3 is a unit of infinite order in ZG, with inverse v = 1 − x − x4.

Let k be the algebraic closure of Fp and K = k(t), where t is an indeterminate. Let G be an elementary p-group of order p^2 generated by a, b. Show that
a → A =
1 1
0 1

b→ B =
1 t
0 1
defines a representation of G over K which is not equivalent to any representation of G over k.

Let Fk(G) be the k-space of class functions on G, given the inner product
[μ, ν] = 1/|G| *(Sum over g) μ(g^−1)ν(g).
Show that, for any class function f ∈ Fk(G), there is a “Fourier expansion” f = (sum over i) [f,χi] χi.

Show that the First Orthogonality Relation can be generalized to
(sum over g∈G) χ_i(g^−1)χ_j(hg) = δ_ij |G|χ_i(h)/n_i,
where h is any element in G, and n_i = χ_i(1).

Let G be a finite group such that, for some field k, kG is a finite direct product of k-division algebras. Show that any subgroup H ⊆ G is normal.

Let k be a field, and H be a normal subgroup of a finite group G. Show that rad kG = kG • rad kH iff char k is not divisor of [G : H].

For any field k and for any normal subgroup H of a group G, show that kH ∩ rad kG = rad kH.